On the determination of the impulsive Sturm–Liouville operator with the eigenparameter‐dependent boundary conditions

Abstract

In the present work, we consider the inverse problem for the impulsive Sturm–Liouville equations with eigenparameter‐dependent boundary conditions on the whole interval (0,π) from interior spectral data. We prove two uniqueness theorems on the potential q(x) and boundary conditions for the interior inverse problem, and using the Weyl function technique, we show that if coefficients of the first boundary condition, that is, h1,h2, are known, then the potential function q(x) and coefficients of the second boundary condition, that is, H1,H2, are uniquely determined by information about the eigenfunctions at the midpoint of the interval and one spectrum or partial information on the eigenfunctions at some internal points and some of two spectra.